Model Checking Linear Logic Specications

ModelChekingLinearLogiSpeiationsM.Bozzano,G.DelzannoandM.MartelliDipartimentodiInformatiaeSienzedell’InformazioneUniversitadiGenovaViaDodeaneso35,16146Genova-Italyfbozzano,giorgio,martelligdisi.unige.itNoInstituteGivenAbstrat.Theoverallgoalofthispaperistoinvestigatethetheoretialfoundationsofalgorithmiveriationtehniquesforspeiationsbasedonrstorderlinearlogi,alogithatanbeusedtonaturallymodelinnitestatesystemswithinternalstrutureddata.ThefragmentweonsiderinthispaperisbasedonthelinearlogiprogramminglanguagealledLO[4℄enrihedwithuniversallyquantiedgoalformulas.AlthoughLOwasoriginallyintroduedasatheoretialfoundationforextensionsoflogiprogramminglanguages,itanalsobeviewedasaverygenerallanguagetospeifyawiderangeofonurrentsystems.OurapproahisbasedontherelationbetweenbakwardreahabilityandprovabilityhighlightedinourpreviousworkonpropositionalLOprograms[9℄.Followingthislineofthoughts,wedenehereageneralframeworkforthebottom-upevaluationofrstorderlinearlogispeiations.Theevaluationproedureisbasedonaneetivexpointoperatorworkingonasymbolirepresentationofinniteolletionsofrstorderlinearlogiformulas.Thetheoryofwell-quasiorderingsanbeusedtoprovidesuÆientonditionsfortheterminationoftheevaluationofnontrivialfragmentsofrstorderlinearlogi.1IntrodutionThealgorithmitehniquesfortheanalysisofPetriNetsarebasedonverywellonsolidatedtheoretialfoundations[19,26,35,20,Finkel,1993,41℄.Severalinterestingproblems,however,beomeundeidablewhenonsideringspeiationlanguagesmoreexpressivethanbasiPetriNets.Inthissetting,validationofomplexspeiationsisoftenperformedthroughsimulationandtesting,i.e.,by\exeutingthespeiationlookingforbugs,e.g.,asinmethodologybasedontheonstrutionofthereahabilitygraphofColoredPetriNets[25℄.Inordertostudyalgorithmitehniquesfortheanalysisofavastrangeofonurrenymodelsitseemsimportanttondauniformframeworktoreasonabouttheirharateristifeatures.Inourapproahwewilladoptlinearlogi[Girard,1987℄asauniedlogialframeworkforonurreny.Asshownin[4,28,34,11,38℄,linearlogiprovidesalogialharaterizationofoneptsandmehanismspeuliarofonurrenylikeloality,reursion,andnondeterminisminthedenitionofaproess[4,28,34℄;ommuniationviasynhronizationandvaluepassing[11,37℄;internalstateandupdatestoitsurrentvalue[4,38℄;andgenerationoffreshnames[12,37℄.Inthissettingprovabilityinfragmentsoflinearlogianbeusedasaformaltooltoreasonaboutbehavioralaspetsoftheonurrentsystems.Theoverallgoalofthispaperistoinvestigatethetheoretialfoundationsofalgorithmiveriationteh-niquesforspeiationsbasedonrstorderlinearlogi,alogithatanbeusedtonaturallymodelinnitestatesystemswithinternalstrutureddata.ThefragmentweonsiderinthispaperisbasedonthelinearlogiprogramminglanguagealledLO[4℄enrihedwithuniversallyquantiedgoalformulas.LOwasoriginallyintroduedasatheoretialfoundationforextensionsoflogiprogramminglanguages.TheappealingfeatureofLO,however,isthatitanalsobeviewedasarihspeiationlanguageforprotoolsandonurrentsystems.Infat,speiationlanguageslikePetriNetsandmultisetrewritinganbenaturallyembeddedintopropositionalLO(seee.g.[4,11,9℄).Furthermore,rstorderLOspeiationsanbeusedtoolortheinternalstateofproesseswithstrutureddatarepresentedasterms,thusenlargingthelassofsystemsthatanbeformallyspeiedinthelogi.Inthisontext,universalquantiationingoalformulashasseveralinterestinginterpretations:itanbeviewedeitherasasortofhidingoperatorinthestyleof-alulus[37℄,orasamehanismtogeneratefreshnamesasin[12℄.Beforedisussinginmoredetailsthetehnialontributionsofourwork,wewillbrieyillustratetheintuitionbehindtheonnetionbetweenPetriNetsandlinearlogi,andbetweenreahabilityandprovabilityintheorrespondingformalsettings.ThebridgebetweenthetwoparadigmsistheproofsasomputationsinterpretationsoflinearlogiproposedbyAndreoliin[3℄.PetriNetsandLinearLogiAPetriNetanberepresentedasamultiset-rewritingsystemoveranitealphabet,sayp;q;r;:::,ofplaenames.Onepossiblewayofexpressingmultisetrewriterulesinlinearlogiisbasedonthefollowingidea.Theonnetive...............................................................................................(multipliativedisjuntion)isinterpretedasamultisetonstrutor,whereastheonnetiveÆ(reversedlinearimpliation)isinterpretedastherewriterelation(seee.g.[4,11,38℄).BothonnetivesareallowedintheLOfragment.Forinstane,asshownin[11℄theLOlausep...............................................................................................qÆp...............................................................................................p...............................................................................................q...............................................................................................tanbeviewedasaPetriNettransitionthatremovesatokenfromplaespandqandputstwotokensinplaep,oneinq,andoneint.Startingfromamultisetofatomiformulas,theappliation(inabakhainingstep[3℄)ofthiskindofrulessimulatestheringofaPetriNettransitionattheorrespondingmarking.Underthisinterpretation,atop-downderivation(i.e.,readfromthegoaltotheaxioms)anbeviewedasarepresentationofoneofthepossib